Introduction
The Compact-to-Dendritic Transition in Reactive Deposition
The first part of the thesis, consisting of chapter 2, investigates the formation of dendrites in reactive deposition. The discrepancy suggests that continuum modeling may be missing a key part of the microscopic dynamics of dendrite formation.
Trajectory Phase Coexistence
As an example of such coexistence, consider a large fluctuation in the average position of the powered walker. Let's say we want to characterize the probability we're measuring as the center of the box 𝑚.
Sampling the Rate Function
As a result, it is not a good choice for exploratory work that may not produce interesting results. Second, cloning techniques are often unknown to researchers outside the field, creating a barrier that prevents the study of large deviations and TPC from being taken up more widely.
The Meaning of a Kink in the SCGF
0 is the typical value of the observable in the reference model and can be obtained from a single trajectory of the reference model. Inspection of the way in which 𝑠 pairs with the rate constants (here any rate involving a hop to the right is multiplied by e−𝑠 and any rate involving a hop to the left is multiplied by e𝑠) reveals that changing 𝑠 moves the reference pattern about the ASEP phase diagram [75, 76].
Compact-to-Dendritic Transition in the Reactive Deposition of
Introduction
From a theoretical point of view, the value of 𝛾 also determines the behavior of the continuum eigenvalue of the discrete model. Finally, for the first time, we also investigate the scaling of the CTD transition in three dimensions.
Model Definitions
The third contribution is that we show that the 𝐷/𝑘 scale of the continuum model, (2.1), has a simple physical interpretation. Specifically, the growth rate of a point on the boundary 𝑣(𝒘) is determined by the flux. 2.8) The constant of proportionality in this equation, 𝜇, is considered small enough that 𝐶 is quasi-constant.
Theory and Methods
We checked the convergence of our time-step simulations by evaluating the critical radius (defined in Section 2.3 ) at different values of the Damköhler number. The ratio between the diffusion constant and the reaction rate constant𝐷/𝑘 plays an important role in the CTD transition.
Results and Discussion
However, by using a larger value of the Damköhler number, we may sacrifice some accuracy with respect to the Da→ 0 limit. First, they imply that the continuum limit of the discrete model, Da → 0, is well defined. Thus, the sharpening of the CTD transition is reminiscent of the finite size scaling of an equilibrium phase transition [35].
And how does the behavior of the transition relate to scaling at finite size in equilibrium.
Conclusion
2.4(b) and (e) show that the CTD transition also becomes sharper in 3D, although we do not yet have enough points to evaluate the prefactor in this dimension. Finally, it remains to be seen how the CTD transition interacts with more complicated depositional geometries. This configuration has been the focus of previous computational and theoretical efforts due to its simplicity [15].
We plan to characterize the CTD transition in this type of deposition in a future publication.
Acknowledgments
Laplacian growth phenomena with a third boundary condition: the transition from a dense structure to a diffusion-limited aggregation fractal. Physical Review A https://link.aps.org/doi/10. Suppression of dendrite formation by pulse charging in rechargeable lithium metal batteries. Journal of Physical Chemistry C. A general term for the growth mode of surface irregularities. Journal of The Electrochemical Society129,2442.
Stability analysis of electrode position over a structured electrolyte with immobilized anions.Journal of The Electrochemical Society161,A847–A855.
Linear Stability Analysis of the Continuum Growth Model
Solving for the second term, ¯𝐶𝐼 𝐼 requires first expanding the reactive boundary condition on the surface to𝑂(𝜖¯𝑚). Consequently, the reactive boundary condition in (2.27) reduces to. 2.31) After expanding each side of this equation to the limit and collecting the terms 𝑂(𝜖¯𝑚), we have for the problem ¯𝐶𝐼 𝐼. Comparing this expression with the boundary equation (2.24), we see that the first term expresses the rate of increase of the radius 𝑣𝜁.
Based on this analysis, Witten and Sander concluded that the critical radius for the compact-to-dendritic transition in the continuum model scales as 𝐷/𝑘 [1].
Brownian Dynamics Simulations
It is created when the incoming supercluster particle bounces off a point within the boundary of another supercluster particle. We set these parameters based on the root-mean-square displacement (RMSD) of the particle per. Given 𝜂 in one dimension, the total RMSD of the particle during a small step is in 𝑑dimensions.
Based on this data point and the trend of bias with Da in 2D, we used the parallel algorithm for our 3D simulations at log10Da =-2.55.
Reactive Deposition on a Lattice
The study of the compact-to-dendritic transition on a lattice raises the question of whether this phenomenon behaves in the same way as it does outside the lattice. Previous authors expected that it would yield that in surface growth models the difference between on-versus off-lattice usually has no effect on the long-length emergent properties of the morphology [ 1 , 16 ]. However, it is now clear that the properties of the compact-to-dendritic transition depend on the underlying geometry of the system.
Specifically, on a lattice, lattice symmetry introduces persistent anisotropy into the dendritic cluster structure [ 17 , 18 ].
First-Hit Distribution in Three Dimensions
Reweighting the reference model trajectories produces an upper bound on the velocity function 𝐽 attached to the original model at the point ˜𝑎. We introduce a reference model (3.7) and (3.9), a variational contribution of the rare dynamics of the original model conditional on 𝑎. For a trajectory𝜔 of the reference model, we want to be able to influence how much.
One additional problem solved by rescaling is the apparent disappearance of the degree function in the limit 𝐿.
Direct evaluation of dynamical large-deviation rate functions
Introduction
𝐽(𝑎) is a degree of deviation function that quantifies the probability of observing certain values of the observed 𝑎 1. This modified or reference model is a microscopic approximation of the behavior of the original model conditioned on certain values of the observed 𝑎. 0, an estimate of the tightness of the boundary and, if the ansatz is well chosen, an exact power function (within statistical error).
Calculating a correction to these limits, by measuring the fluctuations of the likelihood ratio, allows the recovery of the correct rate function.
A variational ansatz for rare dynamics
Evaluating (3.17) from the sample mean of a single trajectory in the reference model yields a point 𝐽. b). Variation of the parameters of the reference model allows reconstruction of the entire blue and (potentially) green curves in panel (a). We want to reweight the reference model trajectories to approximate or calculate𝐽(𝑎) for values of𝑎potentially far from 𝑎0.
1 is large (or cannot be calculated), then it is the very rare trajectories of the reference model that correspond to the desired rare dynamics of the original model.
VARD applied to four examples
The sum (green line) of the limit and the correction (3.18) is equal to the exact rate function, arbitrarily to its end. This limit lies close to the correct answer, even far at the end of the rate function. 55] (dashed gray line); the dotted box in the center of the figure shows the scale of Fig.
In this case, the model is simple enough that each reference model used to calculate the bound performs exactly the sparse dynamics of the original model, conditioned on a particular value of 𝑎.
Conclusions
0we know that the rate function of the reference model vanishes; in contrast, in the equilibrium case we only know the probability of visiting a particular state up to a normalization constant, and we do not know the absolute free energy of the reference model (unless it is particularly simple [80]). VARD provides insight into the approaches used to produce universal rate function bounds from level 2.5 for large deviations, via homogeneous ansätze [54–56], by showing how relaxing such assumptions leads to tightening bounds in different sectors of parameter space. VARD is at the other extreme of the method spectrum in the sense that it only uses a modified dynamic.
8An advantage of bypassing the SCGF and computing the velocity function directly, as we do here, is the ability to reconstruct velocity functions that are not strictly convex.
Acknowledgments
To calculate the probability distribution 𝜌𝑇(𝐴 =𝑎𝑇) of the time-averaged position of the walker, we appeal to the tools of the theory of large deviation. Given the symmetry of the system, the typical location of the walker in the long time limit is 𝑎. 1(𝑥) is the probability that the polymer has position𝑥 on the first row of the lattice.
The steady-state rate function is consistent with the empirical large-deviation rate function of the model in the one-phase and two-phase regions.
Varied phenomenology of models displaying dynamical large-
Introduction
However, singularities with a large deviation do not necessarily imply the existence of cooperative phenomena or distinct phases. We study models of driven random walkers on a lattice that exhibit dynamical singularities of large deviation in the limit of large system size. We also comment on the relationship between ergodic and non-ergodic dynamical systems that exhibit large deviation singularities.
In Section 4.5 we compare the behavior of ergodic and non-ergodic dynamical models representing singularities with large deviations.
Driven random walker
These trajectories show discontinuities, with the walker switching between two locations on either side of the grid. The time frame for residence at different lattice locations increases as 𝐿 increases. e) Current position histograms 𝑥/𝐿 for the trajectories in (d). If the walker sits on any edge of the lattice, it moves away from the edge with probability 1 (so 𝑝(−(𝐿−1)/2) = 1 and 𝑝( (𝐿 analogous to reflection of boundaries in a continuum boundary.
As the grid size 𝐿 increases, SCGF and 𝑎(𝑘) bend more and more sharply and parts of the rate function become more and more linear.
Undriven random walker
The probability of crossing the net in the hard direction is ∼ 𝑝𝐿, and so the time to do so increases exponentially with 𝐿. When the straight lines lie below the blue lines, it is more likely that the system will reach the time average position 𝑎 corresponding to 𝑥/𝐿 discontinuously. This construction is consistent with the conditional trajectories of these models (Fig. 4.5), as long as the trajectory time easily exceeds the time at which the switch occurs.
Hence in this case it is a divergence of the diffusive mixing time 𝐿2 that causes the singularity, not the appearance of discontinuous behavior.
When should we expect intermittency?
Taking into account the different velocities at the edges of the grid, the negative logarithmic probability per unit time is that the walker remains localized near the location. Consistent with the simple arguments developed in this section, the powered walker exhibits discontinuity at most values of 𝑎, involving locations near the edges of the grid. This line lies below the homogeneous result for all𝑥 away from the edges, showing that intermittency is the most likely way to achieve a time average within the grid.
The directed walk shows discontinuity for a wider range of values of 𝑎, and its discontinuity always includes sites near the edges of the lattice.
Singularities in non-ergodic models
Associated with this transition is a change in the shape of the model's large-deviation rate function (see Appendix B) and a dynamical large-deviation singularity. The ensemble nature of the growth model trajectory and walker models is summarized in Figs. Conditional trajectories of delayed progress exhibit discontinuity and a bimodal distribution of the instantaneous coordinate 𝑥/𝐿, but the distribution of the time-integrated quantity is unimodal.
The paths of the upper panels are indicated by dark blue or red lines on the lower panels.
Discussion & Conclusions
The origin of this behavior is cooperativity, the tendency of the model to favor certain behaviors the more behaviors it performs. For the walker models, the SCGF kinks are related to portions of the velocity function that are linear with a zero gradient (the non-powered walker) and with a non-zero gradient (the powered walker). An explicit calculation of the dynamics that give rise to the velocity function is necessary to determine how the model realizes its rare behavior.
All the models discussed here exhibit twists of their dynamic large-deviation functions in certain parameter regimes, but show phase-transition-like behavior to varying degrees.
Acknowledgments
Near the critical point, on either side, the relaxation time of the model is large [15], and empirical rate functions, for the simulated times, depart from the form (B1). 0 = 1000 and composition𝑎, corresponding to the position of the right-hand minimum of the steady rate function (B1). These trajectories populate the middle portions of the empirical rate functions near the critical point.
Consequently, for long times and a large but computationally feasible number of trajectories, the empirical velocity function of the model in the two-phase region is non-convex (concave) and is consistent with the velocity function in the steady state (B1 ).
